In the Shrikhande graph, any two vertices I and J have two distinct neighbors in common (excluding the two vertices I and J themselves), which holds true whether or not I is adjacent to J. In other words, its parameters for being strongly regular are: {16,6,2,2}, with , this equality implying that the graph is associated with a symmetricBIBD. It shares these parameters with a different graph, the 4×4 rook's graph. The 4×4 square grid is the only square grid for which the parameters of the rook's graph are NOT unique, but are shared with a non-rook's graph, namely the Shrikhande graph. [2][3]

The Shrikhande graph is locally hexagonal; that is, the neighbors of each vertex form a cycle of six vertices. As with any locally cyclic graph, the Shrikhande graph is the 1-skeleton of a Whitney triangulation of some surface; in the case of the Shrikhande graph, this surface is a torus in which each vertex is surrounded by six triangles.[4] Thus, the Shrikhande graph is a toroidal graph. The embedding forms a regular map in the torus, with 32 triangular faces. The skeleton of the dual of this map is the Dyck graph, a cubic symmetric graph.